output : keyword option of the form output=value where value is a name or list of names denoting the returned Array image(s). The accepted names are color, or raw. The default value is color.

numterms : keyword option of the form numterms=value where value is a positive integer specifying the number of terms used in the summation that approximates the Lyapunov exponent at each input point The default value is 1000.

container : An n-by-n-by-2 Array with datatype=float[8] and order=Fortran_order used in-place to store the raw data.

The Lyapunov command generates Array images which provide a visualization of a bifurcational fractal that maps regions of chaos and stability given a binary-valued sequence. The particular binary pattern determines the value of r in the iterative formula,

The binary pattern dictates whether r[i] is taken as the x or the y value of a given point in the real plane. The values of z and r are used in computing the Lyapunov exponent according to the following definition.

The above limit is approximated using the finite sum of numterms individual terms. In regions where lambda>0 the behavior is chaotic, while it is stable in regions where lambda<0.

The 2-D grayscale Array image returned by supplying the option output=raw contains the computed values of lambda.

The 3-D color Array image returned by supplying the option output=color contains data where the three layers corresponding to red, green, and blue have been computed using the grayscale raw results.

$\mathrm{with}\left(\mathrm{Fractals}:-\mathrm{EscapeTime}\right)$

$\left[\mathrm{BurningShip}\&comma;\mathrm{Colorize}\&comma;\mathrm{HSVColorize}\&comma;\mathrm{Julia}\&comma;\mathrm{LColorize}\&comma;\mathrm{Lyapunov}\&comma;\mathrm{Mandelbrot}\&comma;\mathrm{Newton}\right]$

$\mathrm{with}\left(\mathrm{ImageTools}\right)\&colon;$

$n≔200\&colon;$

$L≔\mathrm{Lyapunov}\left(n\&comma;\left[1\&comma;0\&comma;0\&comma;1\&comma;1\&comma;0\right]\&comma;2.1..3.9\&comma;2.1..3.9\right)$

$\mathrm{Embed}\left(L\right)$

$L≔\mathrm{Lyapunov}\left(n\&comma;\mathrm{Vector}\left(\left[1\&comma;1\&comma;0\&comma;0\&comma;0\&comma;0\&comma;0\&comma;1\&comma;1\&comma;1\right]\right)\&comma;2.1..3.9\&comma;2.1..3.9\&comma;\mathrm{output}=\mathrm{raw}\right)\&colon;$

$R≔\mathrm{Array}\left(1..n\&comma;1..n\&comma;1..3\&comma;\mathrm{datatype}=\mathrm{float}\left[8\right]\&comma;\mathrm{order}=\mathrm{C\_order}\right)\&colon;$

$\mathrm{LColorize}\left(L\&comma;50\&comma;500\&comma;300\&comma;\mathrm{container}=R\right)\&colon;$

$\mathrm{Embed}\left(\left[\mathrm{FitIntensity}\left(L\right)\&comma;R\right]\right)$

$\mathrm{randomize}\left(\right)\&colon;$

$p≔\mathrm{rand}\left(350..450\right)\&colon;$

$Q≔\left\{\left[0\&comma;0\&comma;0\&comma;0\&comma;1\&comma;1\right]\&comma;\left[0\&comma;0\&comma;1\&comma;0\&comma;0\&comma;1\right]\&comma;\left[0\&comma;1\&comma;0\&comma;0\&comma;0\&comma;1\right]\&comma;\left[0\&comma;1\&comma;1\&comma;0\&comma;0\&comma;1\right]\right\}$

$Q≔\left\{\left[0\&comma;0\&comma;0\&comma;0\&comma;1\&comma;1\right]\&comma;\left[0\&comma;0\&comma;1\&comma;0\&comma;0\&comma;1\right]\&comma;\left[0\&comma;1\&comma;0\&comma;0\&comma;0\&comma;1\right]\&comma;\left[0\&comma;1\&comma;1\&comma;0\&comma;0\&comma;1\right]\right\}$

$L≔\left[\mathrm{seq}\left(\left[\mathrm{seq}\left(\mathrm{LColorize}\left(\mathrm{Complement}\left(\mathrm{Lyapunov}\left(300\&comma;Q\left[i+\left(k-1\right)\cdot 2\right]\&comma;2.1..3.9\&comma;2.1..3.9\&comma;\mathrm{output}=\mathrm{raw}\right)\right)\&comma;p\left(\right)\&comma;2000\&comma;p\left(\right)\right)\&comma;i=1..2\right)\right]\&comma;k=1..2\right)\right]\&colon;$

$\mathrm{Embed}\left(L\right)$

$\mathrm{Embed}\left(\left[\mathrm{seq}\left(\mathrm{Lyapunov}\left(200\&comma;\left[1\&comma;0\&comma;0\&comma;1\&comma;1\&comma;0\right]\&comma;2.1..3.9\&comma;2.1..3.9\&comma;\mathrm{numterms}=i+10\right)\&comma;i=0..600\&comma;200\right)\right]\right)$

The Fractals:-EscapeTime:-Lyapunov command was introduced in Maple 18.

For more information on Maple 18 changes, see Updates in Maple 18.